Local Extrema of Quartic Function: Help Find (x, y) Points

In summary, the function f(x)=2x^4-4x^2+6 has a relative minimum at (-1,4) and (1,4) and a relative maximum at (0,6).
  • #1
MarkFL
Gold Member
MHB
13,288
12
Here is the question:

Can someone please help me find local maximum and local minimum?

Let f(x) = 2x^4 - 4x^2 + 6

Find the point(s) (x, y) at which f achieves:

local maximum - (?,?)
local minimum - (?,?)

can someone help me please
I get 1 and -1 but its wrong apparently

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
  • #2
Hello John,

We are given the function:

\(\displaystyle f(x)=2x^4-4x^2+6\)

And are asked to find the local extrema. In order to do this, we need to equate the first derivative to zero, and finct the critical values:

\(\displaystyle f'(x)=8x^3-8x=8x\left(x^2-1 \right)=8x(x+1)(x-1)=0\)

Hence, our critical values are:

\(\displaystyle x=-1,0,1\)

Observing that the roots here are all off multiplicity 1, and seeing that for $1<x$, we have:

\(\displaystyle f'(x)>0\)

we may then conclude that the sign of the derivative alternates across the 4 intervals made by dividing the real number line at the three critical values, hence we have:

\(\displaystyle f(x)\) increasing on \(\displaystyle (-1,0)\,\cup\,(1,\infty)\)

\(\displaystyle f(x)\) decreasing on \(\displaystyle (-\infty,-1)\,\cup\,(0,1)\)

Thus, by the first derivative test, we are led to conclude that we have the following:

Relative minima at \(\displaystyle \left(-1,f(-1) \right)=(-1,4)\) and \(\displaystyle \left(1,f(1) \right)=(1,4)\)

Relative maximum at \(\displaystyle \left(0,f(0) \right)=(0,6)\)

Here is a plot of the function and its relative extrema:

View attachment 2308
 

Attachments

  • john.jpg
    john.jpg
    14.9 KB · Views: 71

FAQ: Local Extrema of Quartic Function: Help Find (x, y) Points

1. What is a quartic function?

A quartic function is a polynomial function of the form f(x) = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, and e are constants and x is the variable. It is called a quartic function because it has a degree of 4.

2. What are local extrema points?

Local extrema points are points on a graph where the function reaches either a maximum or minimum value within a specific interval. These points are also known as turning points because the graph changes from increasing to decreasing or vice versa at these points.

3. How do you find the local extrema points of a quartic function?

To find the local extrema points of a quartic function, you can use the first derivative test. First, find the derivative of the quartic function. Then, set the derivative equal to 0 and solve for x. The resulting values of x will be the x-coordinates of the local extrema points. To determine whether these points are maximum or minimum points, you can use the second derivative test.

4. Can a quartic function have more than one local extrema point?

Yes, a quartic function can have multiple local extrema points. The number of local extrema points a quartic function has depends on the coefficients of the function and the shape of the graph.

5. How do local extrema points affect the graph of a quartic function?

Local extrema points can affect the shape of the graph of a quartic function. For example, at a maximum point, the graph will change from increasing to decreasing, and at a minimum point, the graph will change from decreasing to increasing. Local extrema points can also indicate important points on the graph, such as the highest or lowest point.

Similar threads

Replies
1
Views
11K
Replies
1
Views
1K
Replies
1
Views
2K
Replies
16
Views
2K
Replies
1
Views
1K
Replies
10
Views
3K
Replies
9
Views
2K
Back
Top